Y^2=z^2-(z-PR)^2 or y=SquareRootOf(z^2-(z-PR)^2 ). If the scissors jack has multiple links (N) then this formula gives the lift from each link so multiply it by N to get the total displacement.
To answer the questions in the other post that is not a linear equation for displacement. Alternatives to scissors jacks include (a) wedges and cams, (b) screws, (c) inflatables like air bags or hydraulic cylinders. The link part of a scissors jack can be employed in a number of ways that seem different that a scissors jack but all use the same type equation for displacement so all are functionally equivalent.